Dirac Structure Research Articles

Radiative Neutron β-Decay in Effective Field Theory.

We consider radiative β-decay of the neutron in heavy baryon chiral perturbation theory. Nucleon-structure effects not encoded in the weak coupling constants gA and gV are determined at next-to-leading order in the chiral expansion, and enter at the -level, making a sensitive test of the Dirac structure of the weak currents possible.

Mirror symmetry and generalized complex manifolds: Part I. The transform on vector bundles, spinors, and branes

In this paper we begin the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇ -semi-flat generalized almost complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇ -semi-flat generalized almost complex structures on the total space of V and ∇ ∨ -semi-flat generalized almost complex structures on the total space of V ∨ . We show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We also study the ways in which our results generalize some aspects of T-duality such as the Buscher rules. We show explicitly how spinors are transformed and discuss the induces correspondence on branes under certain conditions.

Current Algebra and Differential Geometry

We show that symmetries and gauge symmetries of a large class of 2-dimensional sigma models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the sigma model. We compute the current-current commutator and analyse the anomaly cancellation condition, which can be interpreted geometrically in terms of Dirac structures, previously studied in the mathematical literature. Generalized complex structures correspond to decompositions of the current algebra into pairs of anomaly free subalgebras. Sigma models that we can treat with our method include both physical and topological examples, with and without Wess-Zumino type terms.

Geometric modeling of nonlinear RLC circuits

In this paper, the dynamics of nonlinear RLC circuits including independent and controlled voltage or current sources is described using the Brayton-Moser equations. The underlying geometric structure is highlighted and it is shown that the Brayton-Moser equations can be written as a dynamical system with respect to a noncanonical Dirac structure. The state variables are inductor currents and capacitor voltages. The formalism can be extended to include circuits with elements in excess, as well as general noncomplete circuits. Relations with the Hamiltonian formulation of nonlinear electrical circuits are clearly pointed out.

Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Associated with a skew-symmetric linear operator on the spatial domain $[a,b]$ we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinite-dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.

PORT REPRESENTATIONS OF THE TELEGRAPHER'S EQUATIONS

This article studies the telegrapher's equations with boundary port variables. Firstly, a link is made between the telegrapher's equations and a skew-symmetric linear operator on a spatial domain. Associated to this linear operator is a Dirac structure which includes the port variables on the boundary of this spatial domain. Secondly we present all partitions of the port variables into inputs and outputs for which the state dynamics is dissipative. Particularly we recognize the possible input-outputs for which the system is impedance energy-preserving, i.e., as well as scattering energy-preserving, i.e., Additionally, we show how to represent the corresponding system as an abstract infinite-dimensional system, i.e., and.

FROM CONSERVATION LAWS TO PORT-HAMILTONIAN REPRESENTATIONS OF DISTRIBUTED-PARAMETER SYSTEMS

In this paper it is shown how the port-Hamiltonian formulation of distributed-parameter systems is closely related to the general thermodynamic framework of systems of conservation laws and closure equations. The situation turns out to be similar to the lumped-parameter case where the Dirac structure captures the basic interconnection laws, and the closure equations correspond to the constitutive relations of the energy-storing elements.

From Dynamic Graphs to Geometric Interconnection Structures of Physical Systems with Variable Topology

this paper presents a systematic method to obtain a geometric interconnection structure associated with a physical system with variable topology. A non-minimal kernel representation of this hybrid Dirac structure is introduced. This implicit structure includes real and discrete variables and is obtained from a mathematical representation of the dynamic network graph associated with the system. The advantages of this representation are that the discrete state of the switching part is explicit and the equations are directly related to the interconnection constraints, energy supply, storage and dissipation.

Some Aspects of Nuclear Structure in Relativistic Approach**The project supported by National Natural Science Foundation of China under Grant Nos. 10235020 and 10275094 and The State Key Research Development Program under Grant No. G200077407

The nucleon effective interaction in the nuclear medium is investigated in the framework of the Dirac–Brueckner–Hartree–Fock (DBHF) approach. A new decomposition of the Dirac structure of nucleon self-energy in the DBHF is adopted for asymmetric nuclear matter. The properties of finite nuclei are investigated with the nucleon effective interaction. The agreement with the experimental data is satisfactory. The relativistic microscopic optical potential in asymmetric nuclear matter is investigated in the DBHF approach. The proton scattering from nuclei is calculated and compared with the experimental data. A proper treatment of the resonant continuum for exotic nuclei is studied. The width effect of the resonant continuum on the pairing correlation is discussed. The quasiparticle relativistic random phase approximation based on the relativistic mean-field ground state in the response function formalism is also addressed.

Port hamiltonian systems extended to irreversible systems : The example of the heat conduction

In previous work we have proposed a port Hamiltonian formulation of distributed parameter systems with boundary energy flow. These port Hamiltonian systems are defined with respect to a Dirac structure, called Stokes-Dirac structure, in some product space of exterior differential forms. In this paper we show how to extend this formulation to irreversible thermodynamic systems on the example of the heat conduction model. The geometric structure defining the dynamics is shown to be constituted by a Dirac structure constrained on some on its port variables by nonlinear relations associated with the irreversible entropy creation in the system.

Contact manifolds and generalized complex structures

We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.

Dirac–Nijenhuis structures

In this paper we study the concept of Dirac–Nijenhuis structures. We consider them to be a pair where D is a Dirac structure, defined with respect to a Lie bialgebroid (A, A*), and is a Nijenhuis operator which defines a deformation of the Lie algebroid structure of D in a compatible way. The transformation can be considered to deform also the double of the Lie bialgebroid, which leads to the concept of Dirac–Nijenhuis structures of type I, or to not affect it, leading to Dirac–Nijenhuis structures of type II. We prove that the concept contains Poisson–Nijenhuis structures as a particular case and provides another example for the case of Kahler manifolds.

The mathematical representation of driven thermodynamic systems

A general framework for the treatment of driven systems in nonequilibrium thermodynamics is discussed for two selected theories and a simple model system. The framework is based upon the of modeling and control of general physical systems proposed by van der Schaft and co-workers. The crucial concept is the notion of a Dirac structure representing the dynamical equations of motion as well as the power conserving interconnection structure of the system. We applied the framework to two existing theories and a very simple model system. The two selected theories are the “General Equation for the Non-Equilibrium Reversible-Irreversible Coupling” (GENERIC) formalism of Grmela and Öttinger and the Matrix model of Jongschaap; the model system is a viscous gas in a cylinder and an externally driven piston. It is shown that the new approach provides not only a common framework for both theories, but also useful extensions, in particular, an extended GENERIC treatment of driven systems.

Integration of twisted Dirac brackets

The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form. More generally, given a Lie groupoid $G$ over a manifold $M$, we show that multiplicative 2-forms on $G$ relatively closed with respect to a closed 3-form $\phi$ on $M$ correspond to maps from the Lie algebroid of $G$ into the cotangent bundle $T^*M$ of $M$, satisfying an algebraic condition and a differential condition with respect to the $\phi$-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.

Loop corrections to subleading heavy quark currents in SCET

We compute the one-loop (hard) matching correction to heavy-to-light transition currents in soft-collinear effective theory (SCET) to sub-leading power in the SCET expansion parameter for an arbitrary Dirac structure of the QCD weak current.

Radiative neutron β-decay in effective field theory

We consider radiative β-decay of the neutron in heavy baryon chiral perturbation theory, with an extension including explicit Δ degrees of freedom. We compute the photon energy spectrum as well as the photon polarization; both observables are dominated by the electron bremsstrahlung contribution. Nucleon-structure effects not encoded in the weak coupling constants gA and gV are determined at next-to-leading order in the chiral expansion, and enter at the O(0.5%)-level, making a sensitive test of the Dirac structure of the weak currents possible.

Courant algebroid and Lie bialgebroid contractions

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17B62, 53C15, 53D17.

Dirac structures and paracomplex manifolds

We introduce the notion of a generalized paracomplex structure. This is a natural notion which unifies several geometric structures such as symplectic forms, paracomplex structures, and Poisson structures. We show that generalized paracomplex structures are in one-to-one correspondence with pairs of transversal Dirac structures on a smooth manifold. To cite this article: A. Wade, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Singular reduction of implicit Hamiltonian systems

This paper develops the theory of singular reduction for implicit Hamiltonian systems admitting a symmetry Lie group. The reduction is performed at a singular value of the momentum map. This leads to a singular reduced topological space which is not a smooth manifold. A topological Dirac structure on this space is defined in terms of a generalized Poisson bracket and a vector space of derivations, both being defined on a set of smooth functions. A corresponding Hamiltonian formalism is described. It is shown that solutions of the original system descend to solutions of the reduced system. Finally, if the generalized Poisson bracket is nondegenerate, then the singular reduced space can be decomposed into a set of smooth manifolds called pieces. The singular reduced system restricts to a regular reduced implicit Hamiltonian system on each of these pieces. The results in this paper naturally extend the singular reduction theory as previously developed for symplectic or Poisson Hamiltonian systems.

Studying the role of the 't Hooft interaction in QCD, by means of lattice simulations

We report on a recent investigation on the dynamics of light quarks, at intermediate and low energy. By measuring some appropriate correlation functions on the lattice, it is possible to probe the Dirac and flavor structure of the non-perturbative quark-quark interaction and look directly for signatures of the 't Hooft Lagrangian. Results obtained with chiral fermions strongly support the instanton picture and the phenomenological parameters of the Instanton Liquid Model